Fall 2014: Intro-Modern Analysis I
Practice Final Name (Last, First):
0 Answer the questions in the spaces provided on the question sheets.
If you run out of room for an answer, continue on the back of the page.
0 NO calculators or other electro
Analysis I: Solutions to PSET 6
Problem 1
(a) Recall from Rudin Theorem 3.3 that the limit of a product is the product
of the limits if the individual limits exist and hence
n
n
lim n n .
lim 2n = lim 2
n
n
n
From Rudin Theorem 3.20, we nd that limn n 2 =
HW 1
Write and submit:
(1) 2, 3, 5 on page 22,
(2) (a) Suppose S is a subset of real numbers R and R, and let
T = cfw_s + | s S. Show that if S has the least upper bound,
sup S, then T also has the least upper bound, sup T . Moreover,
sup T = sup S + .
(
Solutions to Assignment 9
1. Rudin 5.2
Let a < x < y < b. Then, f is continuous on [x, y] and dierentiable on
(x, y). Thus, by the Mean Value Theorem, there exists c (x, y) such that
f (y) f (x) = (y x)f (c) > 0, where the inequality follows because f (c)
Analysis I: Solutions to PSET 10
Rudin 6.4
For a < b R let f : [a, b] R be 0 for all irrationals and 1 for all rationals.
The density of Q and R \ Q in R force
n
n
U (P, f ) =
sup
xi1 xxi
i=1
n
L(P, f ) =
inf
xi1 xxi
i=1
xi = b a
f (x) xi =
i=1
n
0 xi = 0
Fall 2014: Intro-Modern Analysis I
Practice Exam I
Name (Last, First)/ UNI:
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Fall 2014: Intro-Modern Analysis I
Practice Exam II
Name (Last, First)/ UNI:
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Fall 2014: Intro-Modern Analysis I
Practice Final
Name (Last, First):
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If you run out of room for an answer, continue on the back of the page.
NO calculators or other electronic devices,
ANALYSIS I
9
9.1
The Cauchy Criterion
Cauchys insight
Our diculty in proving an is this: What is ? Cauchy saw that it was enough to
show that if the terms of the sequence got suciently close to each other. then completeness
will guarantee convergence.
Rem
Fall 2013: Intro-Modern Analysis I
Practice Exam II
Name (Last, First):
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MATH 4061, Introduction to Modern Analysis I
Homework 6
Due Monday, October 31st.
Name(s):
UNI(s):
Read carefully the following instructions. You might be penalized for not following them:
(1) Textbook problems refer to W. Rudin, Principles of Mathematica
MATH 4061, Introduction to Modern Analysis I
Homework 5
Due Monday, October 24th.
Name(s):
UNI(s):
Read carefully the following instructions. You might be penalized for not following them:
(1) Textbook problems refer to W. Rudin, Principles of Mathematica
Analysis I: Solutions to PSET 8
Rudin 4.7
The arithmetic/geometric mean inequality shows that f is bounded:
|xy 2 |/2
|xy 2 |
= 2
x2 + y 4
(x + y 4 )/2
|xy 2 |/2
1
= .
2y4
2
x
|f (x, y)| =
However, f is not continuous the limit of f along x = y 2 at (0, 0
Solutions to Assignment 5
1. Rudin 2.15
Dene En := (0, 1/n), n N. Then En is bounded, and for any nite subcollection cfw_Ei iI , let n = miniI cfw_i. Then iI Ei = (0, 1/n) = . Yet we have
that n=1 En = since if 0 < r R we can choose n N such that 1/n < r.
Analysis I: Solutions to PSET 4
Problem 1
Suppose for the sake of contradiction that E F is disconnected, i.e. there
exist opens A, B M such that AB = E F and AB = . Then (AE)
and (B E) provide a separation for E. Since E is connected, we nd, say,
that A
HW 4
Write and submit:
1. If E and F are connected subsets of M with E F = , show that
E F is connected.
2. 16 on page 44.
3. If K is nonempty compact subset of R, show that sup K and inf K are
elements of K.
4. If A is compact in M and B is compact in N
HW 2
Write and submit:
(1) 17 on page 23,
(2) 2 on page 43,
(3) Show that (0, 1) is equivalent to [0, 1] and also equivalent to R.
(4) Let cfw_En , n = 1, 2, 3, be a sequence of countable sets, and put
S = E1 E2 En . Show that S is uncountable. Prove tha
HW 3
Write and submit:
1. 11 on page 44.
2. Check that d(f, g) = maxaxb |f (x)g(x)| denes a metric on C ([a, b]),
the collection of all continuous and real valued functions dened on the
closed interval [a, b].
3. Prove the following two statements:
1
1
1
HW 6
Write and submit:
1. Prove the following two limits.
n2
n
2n = 1, [b.] lim n = 0.
[a.] lim
n 2
n
2. If cfw_xn is a sequence of real numbers and dene its arithmetic means
an by
x0 + x1 + + xn
an =
, n = 0, 1, 2, 3, .
n+1
(a). If lim xn = a, prove th
HW 5
Write and submit:
1. 15 on page 44.
2. 1 on page 78.
3. 3 on page 78.
4. If xn x in (M, d), show that d(xn , y) d(x, y) for any y M .
More generally, if xn x and yn y, show that d(xn , yn ) d(x, y).
Practice:
1. 26 on page 45,
2. A sequence cfw_xn
HW 7
Write and submit:
1. 9 on page 79.
2. 12 on page 79.
3. 1 on page 98.
4. Let f : R R be continuous.
(a). If f (0) > 0, show that f (x) > 0 for all x in some open interval
(a, a).
(b). If f (x) 0 for every rational x, show that f (x) 0 for all real x
HW 10
Write and submit:
1. 4 on page 138,
2. 7 on page 138,
3. 8 on page 138,
4. 10 (a), (b), (c) on page 139. (In Part (c), the problem says that
f and g are complex functions, you just need to consider the case f
and g are real-valued functions. Indeed
HW 8
Write and submit:
1. 7 on page 99,
2. 8 on page 99,
3. 14 on page 100,
4. 18 on page 100.
Practice:
1. Read Theorem 4.20 on page 91 and try to understand the examples
given in the proof.
2. Read Theorem 4.29 and its corollary on page 96.
3. 10 on p
Solutions to Assignment 1
Part (1)
Exercise 2
Assume by contradiction that there exists a rational number m/n with m, n = 0
without common factors such that m2 /n2 = 12, We have m2 = 12n2 . Hence 3
and 2 divide m2 , thus 6 divides m. Call m = 6m1 and then
Analysis I: Solutions to PSET 2
Rudin 1.17
Take two vectors x, y Rk . We claim that
|x + y|2 + |x y|2 = 2|x|2 + 2|y|2 ,
which follows by the commutativity and distributivity of the dot product,
|x + y|2 + |x y|2 = (x + y) (x + y) + (x y) (x y)
= x x + 2x
Solutions to Assignment 3
1. Rudin 2.11
d1
d2
d3
d4
d5
is
is
is
is
is
not a metric, since d1 (2, 0) = 4 > 2 = d1 (2, 1) + d1 (1, 0)
a metric.
not a metric, since d3 (1, 1) = 0
not a metric, since d4 (2, 1) = 0
a metric.
2.
Let f, g, h be continuous and re
MATH 4061, Introduction to Modern Analysis I
Homework 4
Due Monday, October 17th.
Name(s):
UNI(s):
Read carefully the following instructions. You might be penalized for not following them:
(1) Textbook problems refer to W. Rudin, Principles of Mathematica